MathDB

6

Part of 2023 Rioplatense Mathematical Olympiad

Problems(3)

An ant who likes run

Source: Rioplatense L3 2023 #6

12/6/2023
Let ABCABC be an acute-angled triangle such that AB+BC=4ACAB+BC=4AC. Let DD in ACAC such that BDBD is angle bisector of ∠ABC\angle ABC. In the segment BDBD, points PP and QQ are marked such that BP=2DQBP=2DQ. The perpendicular line to BDBD, passing by QQ, cuts the segments ABAB and BCBC in XX and YY, respectively. Let LL be the parallel line to ACAC passing by PP. The point BB is in a different half-plane(with respect to the line LL) of the points XX and YY. An ant starts a run in the point XX, goes to a point in the line ACAC, after that goes to a point in the line LL, returns to a point in the line ACAC and finishes in the point YY. Prove that the least length of the ant's run is equal to 4XY4XY.
geometryangle bisector
FE today, FE tomorrow, FE...

Source: Rioplatense L2 2023 #6

12/6/2023
Find all functions f:Z→Zf:\mathbb{Z} \rightarrow \mathbb{Z} such that f(x+f(y+1))+f(xy)=f(x+1)(f(y)+1)f(x+f(y+1))+f(xy)=f(x+1)(f(y)+1) for any x,yx,y integers.
algebrafunction
Switching computers during a videogame contest

Source: Rioplatense L1 2023 #6

12/7/2023
A group of 40464046 friends will play a videogame tournament. For that, 20232023 of them will go to one room which the computers are labeled with a1,a2,…,a2023a_1,a_2,\dots,a_{2023} and the other 20232023 friends go to another room which the computers are labeled with b1,b2,…,b2023b_1,b_2,\dots,b_{2023}. The player of computer aia_i always challenges the players of computer bi,bi+2,bi+3,bi+4b_i,b_{i+2},b_{i+3},b_{i+4}(the player doesn't challenge bi+1b_{i+1}). After the first round, inside both rooms, the players may switch the computers. After the reordering, all the players realize that they are challenging the same players of the first round. Prove that if one player has not switched his computer, then all the players have not switched their computers.
combinatorics